Is the product of two invertible matrices also invertible?

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Is the product of two invertible matrices also invertible?

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Answer 1 · 2015-10-20T06:30:32

Yes

Explanation:

Matrix multiplication is associative, so $(A B) C = A (B C)$ $(AB)C = A(BC)$ and we can just write $A B C$ $ABC$ unambiguously.

Suppose $A$ $A$ and $B$ $B$ are invertible, with inverses $A^{- 1}$ $A^-1$ and $B^{- 1}$ $B^-1$ . Then $B^{- 1} A^{- 1}$ $B^-1A^-1$ is the inverse of $A B$ $AB$ :

$(A B) (B^{- 1} A^{- 1}) = A B B^{- 1} A^{- 1} = A I A^{- 1} = A A^{- 1} = I$ $(AB)(B^-1A^-1) = ABB^-1A^-1 = AIA^-1 = A A^-1 = I$

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Is the product of two invertible matrices also invertible?

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